3.39 \(\int (-i \sinh (c+d x))^n \, dx\)

Optimal. Leaf size=72 \[ \frac{i \cosh (c+d x) (-i \sinh (c+d x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};-\sinh ^2(c+d x)\right )}{d (n+1) \sqrt{\cosh ^2(c+d x)}} \]

[Out]

(I*Cosh[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, -Sinh[c + d*x]^2]*((-I)*Sinh[c + d*x])^(1 + n))/
(d*(1 + n)*Sqrt[Cosh[c + d*x]^2])

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Rubi [A]  time = 0.0157407, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2643} \[ \frac{i \cosh (c+d x) (-i \sinh (c+d x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};-\sinh ^2(c+d x)\right )}{d (n+1) \sqrt{\cosh ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[((-I)*Sinh[c + d*x])^n,x]

[Out]

(I*Cosh[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, -Sinh[c + d*x]^2]*((-I)*Sinh[c + d*x])^(1 + n))/
(d*(1 + n)*Sqrt[Cosh[c + d*x]^2])

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int (-i \sinh (c+d x))^n \, dx &=\frac{i \cosh (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1+n}{2};\frac{3+n}{2};-\sinh ^2(c+d x)\right ) (-i \sinh (c+d x))^{1+n}}{d (1+n) \sqrt{\cosh ^2(c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0400559, size = 67, normalized size = 0.93 \[ \frac{\sqrt{\cosh ^2(c+d x)} \tanh (c+d x) (-i \sinh (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};-\sinh ^2(c+d x)\right )}{d (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[((-I)*Sinh[c + d*x])^n,x]

[Out]

(Sqrt[Cosh[c + d*x]^2]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, -Sinh[c + d*x]^2]*((-I)*Sinh[c + d*x])^n*T
anh[c + d*x])/(d*(1 + n))

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Maple [F]  time = 0.143, size = 0, normalized size = 0. \begin{align*} \int \left ( -i\sinh \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-I*sinh(d*x+c))^n,x)

[Out]

int((-I*sinh(d*x+c))^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (-i \, \sinh \left (d x + c\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-I*sinh(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((-I*sinh(d*x + c))^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (\frac{1}{2} \,{\left (-i \, e^{\left (2 \, d x + 2 \, c\right )} + i\right )} e^{\left (-d x - c\right )}\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-I*sinh(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((1/2*(-I*e^(2*d*x + 2*c) + I)*e^(-d*x - c))^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- i \sinh{\left (c + d x \right )}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-I*sinh(d*x+c))**n,x)

[Out]

Integral((-I*sinh(c + d*x))**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (-i \, \sinh \left (d x + c\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-I*sinh(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((-I*sinh(d*x + c))^n, x)